Answer
The moment about the $x$-axis is
$$
\begin{aligned}
M_{x}&=\sum_{i=1}^{3} m_{i} y_{i} \\
&=10
\end{aligned}
$$
The moment about the $y$-axis is
$$
\begin{aligned}
M_{y}&=\sum_{i=1}^{3} m_{i} x_{i} \\
&=14
\end{aligned}
$$
The center of mass is
$$
\begin{aligned}
(\overline{x}, \overline{y})& =\left(\frac{M_{y}}{m}, \frac{M_{x}}{m}\right) \\
&=\left(\frac{M_{y}}{\sum_{i=1}^{3} m_{i}}, \frac{M_{x}}{\sum_{i=1}^{3} m_{i}}\right)\\
&=(1.4,1)
\end{aligned}
$$
Work Step by Step
the system:
$$
\begin{array}{l}{m_{1}=4, m_{2}=2, m_{3}=4} \\ {P_{1}(2,-3), P_{2}(-3,1), P_{3}(3,5)}\end{array}
$$
The moment about the $x$-axis is
$$
\begin{aligned}
M_{x}&=\sum_{i=1}^{3} m_{i} y_{i} \\
&=m_{1} y_{1}+m_{2} y_{2}+m_{3} y_{3} \\
&=4\cdot (-3)+2\cdot (1)+4\cdot (5) \\
&=10
\end{aligned}
$$
The moment about the $y$-axis is
$$
\begin{aligned}
M_{y}&=\sum_{i=1}^{3} m_{i} x_{i} \\
&=m_{1} x_{1}+m_{2} x_{2}+m_{3} x_{3} \\
&=4\cdot (2)+2\cdot (-3)+4\cdot (3) \\
&=14
\end{aligned}
$$
The center of mass is
$$
\begin{aligned}
(\overline{x}, \overline{y})& =\left(\frac{M_{y}}{m}, \frac{M_{x}}{m}\right) \\
&=\left(\frac{M_{y}}{\sum_{i=1}^{3} m_{i}}, \frac{M_{x}}{\sum_{i=1}^{3} m_{i}}\right)\\
&=\left(\frac{M_{y}}{m_{1} +m_{2} +m_{3}}, \frac{M_{x}}{m_{1} +m_{2} +m_{3} }\right)\\
&=\left(\frac{M_{y}}{4+2 +4}, \frac{M_{x}}{4+2 +4 }\right)\\
&=\left(\frac{14}{10}, \frac{10}{10}\right) \\
&=(1.4,1)
\end{aligned}
$$
